1. Field of the Invention
The present invention relates generally to methods for modeling the properties of waveguides, and more particularly, to methods for modeling the optical properties of optical waveguides, including but not limited to optical fibers.
2. Description of the Related Art
In a photonic-bandgap fiber (PBF), the cladding is a photonic crystal made of a two-dimensional periodic refractive index structure that creates bandgaps, namely regions in the optical frequency or wavelength domain where propagation is forbidden. This structure can be either a series of concentric rings of alternating high and low indices, as in so-called Bragg fibers (see, e.g., Y. Xu, R. K. Lee, and A. Yariv, Asymptotic analysis of Braggfibers, Optics Letters, Vol. 25, No. 24, Dec. 15, 2000, pages 1756-1758) or a two-dimensional periodic lattice of air holes arranged on a geometric pattern (e.g., triangular), as in air-core PBFs. Examples of air-core PBFs are described by various references, including but not limited to, P. Kaiser and H. W. Astle, Low-loss single material fibers made from pure fused silica, The Bell System Technical Journal, Vol. 53, No. 6, July-August 1974, pages 1021-1039; and J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, Photonic crystal fibers: A new class of optical waveguides, Optical Fiber Technology, Vol. 5, 1999, pages 305-330.
The air hole that forms the core of an air-core PBF is typically larger than the other holes in the PBF and constitutes a defect in the periodic structure of the PBF which introduces guided modes within the previously forbidden bandgap. PBFs offer a number of unique optical properties, including nonlinearities about two orders of magnitude lower than in a conventional fiber, a greater control over the mode properties, and interesting dispersion characteristics, such as modal phase velocities greater than the speed of light in air. For these reasons, in optical communications and sensor applications alike, PBFs offer exciting possibilities previously not available from conventional (solid-core) fibers.
Since the propagation of light in air-core fibers is based on a very different principle than in a conventional fiber, it is important to develop a sound understanding of the still poorly understood properties of air-core fibers, to improve these properties, and to tailor them to specific applications. Since the fairly recent invention of air-core fibers (see, e.g., J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, “Photonic crystals as optical fibers-physics and applications,” Optical Materials, 1999, Vol. 11, pages 143-151), modeling has been an important tool for analyzing the modal properties of air-core PBFs, particularly their phase and group velocity dispersion (see, e.g., B. Kuhlmey, G. Renversez, and D. Maystre, Chromatic dispersion and losses of microstructured optical fibers, Applied Optics, Vol. 42, No. 4, 1 Feb. 2003, pages 634-639; K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion, Optics Express, Vol. 11, No. 8, 21 Apr. 2003, pages 843-852; and G. Renversez, B. Kuhlmey, and R. McPhedran, Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses, Optics Letters, Vol. 28, No. 12, Jun. 15, 2003, pages 989-991), transmission spectra (see, e.g., J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, All-silica single mode optical fiber with photonic crystal cladding, Optics Letters, Vol. 21, No. 19, Oct. 1, 1996, pages 1547-1549; and R. S. Windeler, J. L. Wagener, and D. J. Giovanni, Silica-air microstructured fibers: Properties and applications, Optical Fiber Communications Conference, San Diego, 1999, pages FG1-1 and FG1-2), unique surface modes (see, e.g., D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, Surface modes and loss in air-core photonic band-gap fibers, Proceedings of SPIE Vol. 5000, 2003, pages 161-174; K. Saitoh, N. A. Mortensen, and M. Koshiba, Air-core photonic band-gap fibers: the impact of surface modes, Optics Express, Vol. 12, No. 3, 9 Feb. 2004, pages 394-400; J. A. West, C. M. Smith, N. F. Borelli, D. C. Allan, and K. W. Koch, Surface modes in air-core photonic band-gap fibers, Optics Express, Vol. 12, No. 8, 19 Apr. 2004, pages 1485-1496; M. J. F. Digonnet, H. K. Kim, J. Shin, S. H. Fan, and G. S. Kino, Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers, Optics Express, Vol. 12, No. 9, 3 May 2004, pages 1864-1872; and H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, Designing air-core photonic-bandgap fibers free of surface modes, IEEE Journal of Quantum Electronics, Vol. 40, No. 5, May 2004, pages 551-556), and propagation loss (see, e.g., T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botton, and M. J. Steel, Confinement losses in microstructured optical fibers, Optics Letters, Vol. 26, No. 21, Nov. 1, 2001, pages 1660-1662; D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, Leakage properties of photonic crystal fibers, Optics Express, Vol. 10, No. 23, 18 Nov. 2002, pages 1314-1319; and K. Saitoh and M. Koshiba, Leakage loss and group velocity dispersion in air-core photonic bandgap fibers, Optics Express, Vol. 11, No. 23, 17 Nov. 2003, pages 3100-3109). However, extensive and systematic parametric studies of these fibers are lacking because simulations of these complex structures are conducted with analytical methods that are either time-consuming, or are not applicable to all PBF geometries. For example, simulating all the modes of a typically single-mode or a few-moded air-core PBF using code currently in use, for example, the MIT Photonic-Band (MPB) code for computing dispersion relations and electromagnetic profiles of the modes of periodic dielectric structures, takes approximately 10 hours on a supercomputer using 16 parallel 2-GHz processors, with a cell size of 10Λ×10Λ and a spatial resolution of Λ/16, where Λ is the period of the photonic-crystal cladding. (See, e.g., S. G. Johnson, and J. D. Joannopoulos, Block-iterative frequency-domain methods for Maxwell's equations in planewave basis, Optics Express, Vol. 8, No. 3, 29 Jan. 2001, pages 173-190 for further information regarding the MPB code.) Faster computations have been reported using a polar-coordinate decomposition method to model the fundamental mode (effective index and fields) in six minutes (see, e.g., L. Poladian, N. A. Issa, and T. M. Monro, “Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry,” Optics Express, 2002, Vol. 10, pages 449-454). However, this high-speed was accomplished by limiting the number of air hole layers to one, and it still required a supercomputer.
In addition, a multipole decomposition method (see, e.g., T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith and C. Martijn de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Optics Express, 2001, Vol. 8, pages 721-732) has been able to model all the modes in a hexagonal PBF with four rings of holes at one wavelength in approximately one hour on a 733-MHz personal computer. This figure would be reduced to a few minutes on a faster personal computer (e.g., approximately 15 minutes on a 3.2-GHz processor.) While this multipole method yields accurate and quick calculations of the modes of PBFs with circular or elliptical hole shapes, it is limited to these shapes and does not work for PBFs with other hole shapes, for example, for PBFs with scalloped cores (which can advantageously avoid undesirable surface modes) or nearly-hexagonal cladding holes (a shape which often occurs in actual air-core fibers). In addition, much finer resolutions are needed to accurately model the extremely small physical features of air-core fibers, in particular their very thin membranes (e.g., often thinner than Λ/100).
PBFs are typically modeled using numerical methods originally developed for the study of photonic crystals. Most methods are based on solving the photonic-crystal master equation:
                              ∇                      ×                          {                                                1                                      ɛ                    ⁡                                          (                      r                      )                                                                      ⁢                                  ∇                                      ×                                          H                      ⁡                                              (                        r                        )                                                                                                        }                                      =                                            ω              2                                      c              2                                ⁢                      H            ⁡                          (              r              )                                                          (        1        )            where r is a vector (x, y) that represents the coordinates of a particular point in a plane perpendicular to the fiber axis, ∈(r)=n2(r) is the dielectric permeability of the fiber cross-section at this point, where n(r) is the two-dimensional refractive index profile of the fiber, H(r) is the mode's magnetic field vector, ω is the optical frequency; and c is the speed of light in vacuum. This formulation is most useful for three-dimensional (3D) photonic-bandgap structures, and it is solved by assuming a constant value for the propagation constant kz and computing the eigenfrequencies ω for this value.
Various methods have previously been used to solve for modes in the context of air-core PBFs. One previously-used method utilizes the MIT Photonic Bandgap (MPB) code or software which is cited above. Using the MPB code, the solutions of Equation 1 are obtained in the spatial Fourier domain, so the first step is to decompose the mode fields and 1/∈(r) in spatial harmonics. Equation 1 is then written in a matrix form and solved by finding the eigenvalues of this matrix. This calculation is done by setting the wavenumber kz to some value (e.g., k0) and by solving for the frequencies at which a mode with this wavenumber occurs.
Another previously-used method is the beam-propagation method (BPM) (see, e.g., M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Optics Technology Letters, 2001, Vol. 30, pages 327-330). In the BPM, a pseudo-random field is launched into the fiber, which excites different modes with different effective indices. The propagation axis is transformed into an imaginary axis, so that the effective indices become purely imaginary and the fiber modes become purely lossy. Instead of accumulating phase at different rates, the modes now attenuate at different rates, which allow the BPM method to separate them and extract them iteratively by propagating them through a length of fiber. The mode with the lowest effective index, which attenuates at the smallest rate, is extracted first, and the other modes are computed in order of increasing effective index.
Another previously-used method is the finite domain time difference method (FDTD), which solves the time-dependent Maxwell's equations:
                    {                                                                              ∇                                      ×                    E                                                  =                                                      -                                          μ                      0                                                        ⁢                                                            ∂                      H                                                              ∂                      t                                                                                                                                                                ∇                                      ×                    H                                                  =                                                      ɛ                    0                                    ⁢                                      ɛ                    ⁡                                          (                      r                      )                                                        ⁢                                                            ∂                      E                                                              ∂                      t                                                                                                                              (        2        )            Solving these equations requires considering the time variable as well (see, e.g., J. M. Pottage, D. M. Bird, T. D. Hedley, T. A. Birks, J. C. Knight, P. J. Roberts, and P. St. J. Russell, Robust photonic band gaps for hollow core guidance in PBF made from high index glass, Optics Express, Vol. 11, No. 22, 3 Nov. 2003, pages 2854-2861).
Another previously-used method involves using a two-dimensional mode eigenvalue equation, where the eigenvalue is the mode effective index. The mode is then expanded either in plane-waves (see, e.g., K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE Journal of Quantum Electronics, 2002, Vol. 38, pages 927-933) or in wavelet functions (see, e.g., W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, Supercell lattice method for photonic crystal fibers, Optics Express, Vol. 11, No. 9, 5 May 2003, pages 980-991). The refractive index profile is decomposed in Fourier components, and the resulting matrix equation is solved for eigenvalues and eigenvectors.
When applied to the calculation of the modes of a PBF, the foregoing previously-used methods suffer from a number of disadvantages. One disadvantage is that computation times can be extremely long. Another disadvantage is that techniques that rely on a mode expansion require calculating the Fourier transform of the refractive index profile of the fiber. For PBFs and other types of waveguides, the refractive index profile contains very fine features (e.g., thin membranes) and abrupt refractive index discontinuities (e.g., at the air-glass interfaces), both of which are key to the mode behavior, so a large number of harmonics should ideally be retained to faithfully describe these features. However, because retaining these large numbers of harmonics would require much more memory and computer processing power than is available, even in supercomputers, the discontinuities in the refractive index profile are smoothened, which artificially widens the fine features (e.g., membranes). This standard practice results in significant differences between the refractive index profile that is being modeled and the actual refractive index profile, which in turn introduces systematic errors in the calculations.
Another disadvantage is that some existing mode solvers must calculate the modes of the highest order bands first and work their way down to lower-order modes, and this process propagates computation errors. This requirement also increases computation time, since it forces the computer to calculate more modes than the ones of interest. In addition, as described above, the multipole decomposition method only works for a few specific hole shapes (e.g., circular and elliptical), thus making this technique less versatile.